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Homework Help: Limits of Integration and finding k

  1. Nov 29, 2008 #1
    1. The problem statement, all variables and given/known data

    Joint pdf given as kxy for 0 < x < y < 1.

    Find the value of k.



    3. The attempt at a solution

    I understand the process of finding k - doing the double integral and setting it to 1. What I dont understand is the limits of integration for y.

    I've seen two different limits set, but I still cannot seem to figure out how and why it is done.

    I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y. Both give the correct answer of k = 8. My question is how do you go about choosing the limits for the y integral?

    Thanks a lot!
     
  2. jcsd
  3. Nov 29, 2008 #2
    The latter does not make sense; it more likely read "integral of y from 0 to 1 and the integral of x from 0 to y".

    The region being integrated over is the triangle with vertices (0,0), (1,1) and (0,1). Using vertical strips, we have the limits

    [tex]\int_{x \, = \, 0}^{x=1} \int_{y=x}^{y=1} kxy \, dy \, dx[/tex],

    or, using horizontal strips, we have

    [tex]\int_{y=0}^{y=1} \int_{x=0}^{x=y} kxy \, dx \, dy[/tex].

    (The limits in the integrals read "x=0", "x=1" etc -- the equals signs look like minus signs in this latex, unfortunately; I've included them for clarity.)

    Both are equivalent.
     
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